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Formation and construction of a shock wave for 3-D compressible Euler equations with the spherical initial data

Published online by Cambridge University Press:  22 January 2016

Huicheng Yin
Huicheng Yin*
Affiliation:
Department of Mathematics and IMS, Nanjing University, Nanjing 210093, P. R. China, huicheng@nju.edu.cn
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Abstract

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In this paper, the problem on formation and construction of a shock wave for three dimensional compressible Euler equations with the small perturbed spherical initial data is studied. If the given smooth initial data satisfy certain nondegeneracy conditions, then from the results in [22], we know that there exists a unique blowup point at the blowup time such that the first order derivatives of a smooth solution blow up, while the solution itself is still continuous at the blowup point. From the blowup point, we construct a weak entropy solution which is not uniformly Lipschitz continuous on two sides of a shock curve. Moreover the strength of the constructed shock is zero at the blowup point and then gradually increases. Additionally, some detailed and precise estimates on the solution are obtained in a neighbourhood of the blowup point.

Keywords

35L7035L65
Type
Research Article
Information
Nagoya Mathematical Journal , Volume 175 , 2004 , pp. 125 - 164
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 2004

References

[1] Alinhac, S., Temps de vie precise et explosion geometrique pour des systemes hyperboliques quasilineaires en dimension un d’espace, Ann. Scoula Norm. Sup. Pisa. Serie IV, XXII (1995), no. 3, 493–515.Google Scholar
[2] Alinhac, S., Blowup of small data solutions for a class of quasilinear wave equations in two space dimensions, Ann. of Math., 149 (1999), no. 1, 97–127.CrossRefGoogle Scholar
[3] Bressan, A., Crasta, G. and Piccoli, B., Well-posedness of the Cauchy problem for n × n systems of conservation laws, Mem. Amer. Math. Soc. 146, no. 694, 2000.CrossRefGoogle Scholar
[4] Chen, S. and Dong, L., Formation of shock for the p-system with general smooth initial data, Sci. in China, Ser. A, 44 (2001), no. 9, 1139–1147.CrossRefGoogle Scholar
[5] Chen, S., Xin, Z. and Yin, H., Formation and construction of shock wave for quasilinear hyperbolic system and its application to 1-D inviscid compressible flow, preprint (1999).Google Scholar
[6] Chen, S. and Zhang, Z. B., On the generation of shock waves of first order quasilinear equations, Fudan Journal (Natural Science) (1963), 13–22.Google Scholar
[7] Chen, G. Q. and Glimm, J., Global solutions to the compressible Euler equations with geometrical structure, Comm. Math. Phys., 180 (1996), 153–193.CrossRefGoogle Scholar
[8] Dafermos, C. M., Generalized characteristics in hyperbolic system of conservation laws, Arch. Rat. Mech. Anal., 107 (1989), 127–155.CrossRefGoogle Scholar
[9] Diperna, R., Uniqueness of solutions to hyperbolic conservation laws, Indiana Univ. Math. J., 28 (1979), 224–257.CrossRefGoogle Scholar
[10] Glimm, J., Solution in the large for nonlinear hyperbolic systems of equations, Comm. Pure Appl. Math., 18 (1965), 697–715.CrossRefGoogle Scholar
[11] Hömander, L., Lectures on nonlinear hyperbolic differential equations, Mathematics and Applications 26, Springer-Verlag, Berlin, 1997.Google Scholar
[12] John, F., Formation of singularities in one-dimensional nonlinear wave propagation, Comm. Pure Appl. Math., 27 (1974), 377–405.CrossRefGoogle Scholar
[13] John, F., Existence for large times of strict solutions of nonlinear wave equations in three space for small initial data, Comm. Pure Appl. Math., 40 (1987), no. 1, 79–109.CrossRefGoogle Scholar
[14] John, F. and Klainerman, S., Almost global existence to nonlinear wave equations in three space, Comm. Pure Appl. Math., 37 (1984), no. 4, 443–455.CrossRefGoogle Scholar
[15] Lax, P. D., Hyperbolic systems of conservation laws II, Comm. Pure Appl. Math., X (1957), 537–566.CrossRefGoogle Scholar
[16] Lax, P. D., Hyperbolic systems of conservation laws and the mathematical theory of shocks waves, Conf. Board Math. Sci. SIAM, 11, 1973.Google Scholar
[17] Lebaud, M. P., Description de la formation d’un choc dans le p-systems, J. Math. Pures Appl., 73 (1994), 523–565.Google Scholar
[18] Liu, T., Development of singularities in the nonlinear waves for quasilinear hyperbolic partial differential equations, J. D. E., 33 (1979), 92–111.CrossRefGoogle Scholar
[19] Liu, T. P. and Yong, T., Well-posedness theory for hyperbolic conservation laws, Comm. Pure Appl. Math., 52 (1999), no. 12, 1553–1586.3.0.CO;2-S>CrossRefGoogle Scholar
[20] Majda, A., Compressible fluid flow and systems of conservation laws in several space variables, Springer-Verlag, 1984.CrossRefGoogle Scholar
[21] Smoller, J. A., Shock waves and reaction-diffusion equations, Berlin-Heiderberg-New York, Springer-Verlag, New York, 1984.Google Scholar
[22] Yin, H., The blowup mechanism of axisymmetric solutions for three dimensional quasilinear wave equations with small data, Science in China (Series A), 43 (2000), no. 3, 252–266.CrossRefGoogle Scholar
[23] Yin, H., The blowup mechanism of small data solutions for the quasilinear wave equations in three space dimensions, Acta Math. Sinica, English Series, 17 (2001), no. 1, 35–76.CrossRefGoogle Scholar
[24] Yin, H. and Qiu, Q., The blowup of solutions for three dimensional spherically symmetric compressible Euler equations, Nagoya M. J., 154 (1999), 157–169.Google Scholar